Coordinate Patch Research Articles

Supersymmetric backgrounds from 5d N $$ \mathcal{N} $$ = 1 supergravity

We construct curved backgrounds with Euclidean signature admitting rigid supersymmetry by using a 5d N=1 off-shell Poincare supergravity. We solve the conditions for the background Weyl multiplet and vector multiplets that preserve at least one supersymmetry parameterized by a symplectic Majorana spinor, and represent the solution in terms of several independent fields. We also show that the partition function does not depends on the local degrees of freedom of the background fields. Namely, as far as we focus on a single coordinate patch, we can freely change the independent fields by combining Q-exact deformations and gauge transformations. We also discuss realization of several known examples of supersymmetric theories in curved backgrounds by using the supergravity.

GENASIS: GENERAL ASTROPHYSICAL SIMULATION SYSTEM. I. REFINABLE MESH AND NONRELATIVISTIC HYDRODYNAMICS

GenASiS (General Astrophysical Simulation System) is a new code being developed initially and primarily, though by no means exclusively, for the simulation of core-collapse supernovae on the world's leading capability supercomputers. This paper---the first in a series---demonstrates a centrally refined coordinate patch suitable for gravitational collapse and documents methods for compressible nonrelativistic hydrodynamics. We benchmark the hydrodynamics capabilities of GenASiS against many standard test problems; the results illustrate the basic competence of our implementation, demonstrate the strengths and limitations of the HLLC relative to the HLL Riemann solver in a number of interesting cases, and provide preliminary indications of the code's ability to scale and to function with cell-by-cell fixed-mesh refinement.

Charting an Inflationary Landscape with Random Matrix Theory

We construct a class of random potentials for N >> 1 scalar fields using non-equilibrium random matrix theory, and then characterize multifield inflation in this setting. By stipulating that the Hessian matrices in adjacent coordinate patches are related by Dyson Brownian motion, we define the potential in the vicinity of a trajectory. This method remains computationally efficient at large N, permitting us to study much larger systems than has been possible with other constructions. We illustrate the utility of our approach with a numerical study of inflation in systems with up to 100 coupled scalar fields. A significant finding is that eigenvalue repulsion sharply reduces the duration of inflation near a critical point of the potential: even if the curvature of the potential is fine-tuned to be small at the critical point, small cross-couplings in the Hessian cause the curvature to grow in the neighborhood of the critical point.

Propagators in de Sitter space

In a spacetime with no global timelike Killing vector, we do not have a natural choice for the vacuum state of matter fields, leading to an ambiguity in defining the Feynman propagators. In this paper, taking the vacuum state to be the instantaneous ground state of the Hamiltonian at each moment, we develop a method for calculating wave functions associated with the vacuum and the corresponding in-in and in-out propagators. We apply this method to free scalar field theory in de Sitter space and obtain de Sitter invariant propagators in various coordinate patches. We show that the in-out propagator in the Poincare patch has a finite massless limit in a de Sitter invariant form. We argue and numerically check that our in-out propagators agree with those obtained by a path integral with the standard i\epsilon prescription, and identify the condition on a foliation of spacetime under which such coincidence can happen for the foliation. We also show that the in-out propagators satisfy Polyakov's composition law. Several applications of our framework are also discussed.

Towards an anomaly-free quantum dynamics for a weak coupling limit of Euclidean gravity: Diffeomorphism covariance

The G-->0 limit of Euclidean gravity introduced by Smolin is described by a generally covariant U(1)xU(1)xU(1) gauge theory. In an earlier paper, Tomlin and Varadarajan constructed the quantum Hamiltonian constraint of density weight 4/3 for this U(1)xU(1)xU(1) theory so as to produce a non-trivial anomaly free LQG-type representation of the Poisson bracket between a pair of Hamiltonian constraints. These constructions involved a choice of regulating coordinate patches. The use of these coordinate patches is in apparent conflict with spatial diffeomorphism covariance. In this work we show how an appropriate choice of coordinate patches together with suitable modifications of these constructions results in the diffeomorphism covariance of the continuum limit action of the Hamiltonian constraint operator, while preserving the anomaly free property of the continuum limit action of its commutator.

The wave equation on the extreme Reissner–Nordström black hole

We study the scalar wave equation on the open exterior region of an extreme Reissner–Nordström black hole and prove that, given compactly supported data on a Cauchy surface orthogonal to the timelike Killing vector field, the solution, together with its (t, s, θ, ϕ) derivatives of arbitrary order, s a tortoise radial coordinate, is bounded by a constant that depends only on the initial data. Our technique does not allow studying transverse derivatives at the horizon, which is outside the coordinate patch that we use. However, using previous results that show that second and higher transverse derivatives at the horizon of a generic solution grow unbounded along horizon generators, we show that any such divergence, if present, would be milder for solutions with compact initial data.

Spacetime splitting, admissible coordinates, and causality

To confront relativity theory with observation, it is necessary to split spacetime into its temporal and spatial components. The (1+3) timelike threading approach involves restrictions on the gravitational potentials $(g_{\mu \nu})$, while the (3+1) spacelike slicing approach involves restrictions on $(g^{\mu \nu})$. These latter coordinate conditions protect chronology within any such coordinate patch. While the threading coordinate conditions can be naturally integrated into the structure of Lorentzian geometry and constitute the standard coordinate conditions in general relativity, this circumstance does not extend to the slicing coordinate conditions. We explore the influence of chronology violation on wave motion. In particular, we consider the propagation of radiation parallel to the rotation axis of stationary G\"odel-type universes characterized by parameters $\eta > 0$ and $\lambda > 0$ such that for $\eta < 1$ ($\eta >1$) chronology is protected (violated). We show that in the WKB approximation such waves can freely propagate only when chronology is protected.

New code for equilibriums and quasiequilibrium initial data of compact objects

We present a new code, named COCAL - Compact Object CALculator, for the computation of equilibriums and quasiequilibrium initial data sets of single or binary compact objects of all kinds. In the cocal code, those solutions are calculated on one or multiple spherical coordinate patches covering the initial hypersurface up to the asymptotic region. The numerical method used to solve field equations written in elliptic form is an adaptation of self-consistent field iterations in which Green's integral formula is computed using multipole expansions and standard finite difference schemes. We extended the method so that it can be used on a computational domain with excised regions for a black hole and a binary companion. Green's functions are constructed for various types of boundary conditions imposed at the surface of the excised regions for black holes. The numerical methods used in cocal are chosen to make the code simpler than any other recent initial data codes, accepting the second order accuracy for the finite difference schemes. We perform convergence tests for time symmetric single black hole data on a single coordinate patch, and binary black hole data on multiple patches. Then, we apply the code to obtain spatially conformally flat binary black hole initial data using boundary conditions including the one based on the existence of equilibrium apparent horizons.

Towards a supersymmetric generalization of the Schwarzschild-(anti)de Sitter space-times

The Wheeler-DeWitt (WDW) equation for the Lambda-Kantowski-Sachs model can also be understood as the WDW-equation corresponding to the Schwarzschild-(anti)deSitter space-times, due to the well-known diffeomorphism between these two metrics. This equation ignores the coordinate patch one chooses and only by imposing coordinate conditions will it be possible to distinguish between black hole or cosmological models. At that point, the foliation parameter $t$ or $r$ will appear in the solution of interest. In this work we supersymmetrize this WDW-equation obtaining an extra term in the potential with two possible signs. We have already, in a similar manner, proposed and analyzed a supersymmetric generalization of a Schwarzschild black hole. The WKB method is then applied, given rise to four classical equations, two for the Schwarzschild-deSitter space-time and another two for the Schwarzschild-anti-deSitter space-time. One can then study the asymptotic cases in which one of two potential terms arising in each Hamiltonian, for each of these space-times, dominates the behavior. One of these limiting (bosonic) cases gives the usual Schwarzschild-(anti)deSitter space-times. We will study here the other four asymptotic regions; they provide six solutions. For the Schwarzschild-deSitter space-time we get two solutions which have singularities at $r=0$ and ${r}_{0}^{s}$, and depending on an integration constant $C$ and the sign at the potential due to these SUSY region, they can also present another two singularities in ${r}_{h}^{s}$ and ${r}_{c}^{s}$. For the Schwarzschild-anti-deSitter space-time the solutions have a singularity at $r=0$ and depending on the integration constant and the sign of the SUSY potential another singularity can appear at ${r}_{+}$. We find associated masses and, based on the holographic principle, we find also the entropies for the bosonic region, which coincide with the ones obtained by the usual methods. We apply this same procedure to get entropies associated to the supersymmetric asymptotic solutions. Even though we were unable to obtain the complete solution to the model, it is shown that horizons can never be reached because when one would approach the standard horizons (the bosonic region), the relevant term in the potential is the one due to supersymmetry and as mentioned in this asymptotic limit one does not have horizons but instead singularities.

High accuracy binary black hole simulations with an extended wave zone

We present results from a new code for binary black hole evolutions using the moving-puncture approach, implementing finite differences in generalized coordinates, and allowing the spacetime to be covered with multiple communicating nonsingular coordinate patches. Here we consider a regular Cartesian near-zone, with adapted spherical grids covering the wave zone. The efficiencies resulting from the use of adapted coordinates allow us to maintain sufficient grid resolution to an artificial outer boundary location which is causally disconnected from the measurement. For the well-studied test case of the inspiral of an equal-mass nonspinning binary (evolved for more than 8 orbits before merger), we determine the phase and amplitude to numerical accuracies better than 0.010% and 0.090% during inspiral, respectively, and 0.003% and 0.153% during merger. The waveforms, including the resolved higher harmonics, are convergent and can be consistently extrapolated to $r\ensuremath{\rightarrow}\ensuremath{\infty}$ throughout the simulation, including the merger and ringdown. Ringdown frequencies for these modes (to $(\ensuremath{\ell},m)=(6,6)$) match perturbative calculations to within 0.01%, providing a strong confirmation that the remnant settles to a Kerr black hole with irreducible mass ${M}_{\mathrm{irr}}=0.884355\ifmmode\pm\else\textpm\fi{}20\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}6}$ and spin ${S}_{f}/{M}_{f}^{2}=0.686923\ifmmode\pm\else\textpm\fi{}10\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}6}$.

Finite Frame Varieties: Nonsingular Points, Tangent Spaces, and Explicit Local Parameterizations

The (μ,S)-frames are frames with lengths in [μ 1⋅⋅⋅μ N ] and with frame operator S, or the \(F=[f_{1}\cdots f_{N}]\in M_{d\times N}(\mathbb{E})\) with column lengths listed by μ and which satisfy FF ∗=S. In this paper, we characterize the nonsingular points of real and complex finite (μ,S)-frame varieties by determining where generalized tori and distorted Stiefel manifolds intersect transversally in Hilbert-Schmidt spheres. This provides us with a characterization of the tangent space for each nonsingular point of the (μ,S)-frame varieties, and we leverage this characterization to demonstrate the existence of structured, locally well defined analytic coordinate patches. We conclude by deriving explicit expressions for these coordinates.

Gravitational Collapse in Painlevé-Gullstrand Coordinates

We construct an exact solution for the spherical gravitational collapse in a single coordinate patch. To describe the dynamics of collapse, we use a generalized form of the Painlev\'e-Gullstrand coordinates in the Schwarzschild spacetime. The time coordinate of the form is the proper time of a free-falling observer so that we can describe the collapsing star not only outside but also inside the event horizon in a single coordinate patch. We show the both solutions corresponding to the gravitational collapse from infinity and from a finite radius.

Grassmannian spectral shooting

We present a new numerical method for computing the pure-point spectrum associated with the linear stability of coherent structures. In the context of the Evans function shooting and matching approach, all the relevant information is carried by the flow projected onto the underlying Grassmann manifold. We show how to numerically construct this projected flow in a stable and robust manner. In particular, the method avoids representation singularities by, in practice, choosing the best coordinate patch representation for the flow as it evolves. The method is analytic in the spectral parameter and of complexity bounded by the order of the spectral problem cubed. For large systems it represents a competitive method to those recently developed that are based on continuous orthogonalization. We demonstrate this by comparing the two methods in three applications: Boussinesq solitary waves, autocatalytic travelling waves and the Ekman boundary layer.

Towards a supersymmetric generalization of the Schwarzschild black hole

The Wheeler-DeWitt (WDW) equation for the Kantowski-Sachs model can also be understood as the WDW-equation corresponding to the Schwarzschild black hole due to the well known diffeomorphism between these two metrics. The WDW-equation and its solutions are ``ignorant'' of the coordinate patch one is using, only by imposing coordinate conditions we can differentiate between cosmological and black hole models. At that point, the foliation parameter $t$ or $r$ will appear in the solution of interest. In this work we supersymmetrize this WDW-equation obtaining an extra term in the potential with two possible signs. The WKB method is then applied, given rise to two classical equations. It is shown that the event horizon can never be reached because, very near to it the extra term in the potential, for each one of the equations, is more relevant than the one that corresponds to Schwarzschild. One can then study the asymptotic cases in which one of the two terms in the Hamiltonian dominates the behavior. One of them corresponds to the usual Schwarzschild black hole. We will study here the other two asymptotic regions; they provide three solutions. All of them have a singularity in $r=0$ and depending on an integration constant $C$ they can also present a singularity in $r=C^2$. Neither of these solutions have a Newtonian limit. The black hole solution we study is analyzed between the singularity $r=C^2$ and a maximum radius $r_m$. We find an associated mass, considering the related cosmological solution inside $r=C^2$, and based on the holographic principle an entropy can be assigned to this asymptotic solution.

Supertubes in bubbling backgrounds: Born-Infeld meets supergravity

We discuss two ways in which one can study two-charge supertubes as components of generic three-charge, three-dipole charge supergravity solutions. The first is using the Born-Infeld action of the supertubes, and the second is via the complete supergravity solution. Even though the Born-Infeld description is only a probe approximation, we find that it gives exactly the same essential physics as the complete supergravity solution. Since supertubes can depend on arbitrary functions, our analysis strengthens the evidence for the existence of three-charge black-hole microstate geometries that depend on an infinite set of parameters, and sets the stage for the computation of the entropy of these backgrounds. We examine numerous other aspects of supertubes in three-charge, three-dipole charge supergravity backgrounds, including chronology protection during mergers, the contribution of supertubes to the charges and angular momenta, and the enhancement of their entropy. In particular, we find that entropy enhancement affects supertube fluctuations both along the internal and the spacetime directions, and we prove that the charges that give the enhanced entropy can be much larger than the asymptotic charges of the solution. We also re-examine the embedding of five-dimensional black rings in Taub-NUT, and show that in different coordinate patches a ring can correspond to different four-dimensional black holes. Last, but not least, we show that all the three-charge black hole microstate geometries constructed so far can be embedded in AdS_3 x S^3, and hence can be related to states of the D1-D5 CFT.

On shallow shell equations

We present explicit formulas for the shallow shell model consisting of a couple of a wave equation and a plate equation, where the middle surface is viewed as a natural manifold with the induced metric from the classical Euclidean space of three dimensions. The Green formula for the shallow is given by the displacement field which expresses the relationship between the interior and the boundary. Next, the ellipticity of the strain energy for the shallow shell is studied under some curvature assumptions on the middle surface. Finally, the motion equations for shallow shells are obtained in terms of the displacement field as an unknown. The new ingredients in these formulas are that they take a form which is not described by a coordinate patch to provide the shell theory with the modern geometry.

NONSTANDARD SEPARABILITY ON THE MINKOWSKI PLANE

We present examples of nonstandard separation of the natural Hamilton–Jacobi equation on the Minkowski plane 2. By "nonstandard" we refer to the cases in which the form of the metric, when expressed in separating coordinates, does not have the usual Liouville structure. There are two possibilities: the "complex-Liouville" (or "harmonic") case and the "linear/null" (or "Jordan block") case. By means of explicit examples, we show that, in all cases, a suitable glueing of coordinate patches of the different structures allows us to separate natural systems with indefinite kinetic energy all over 2.

Comments on Bona–Massó-type slicing conditions in long-term black hole evolutions

We review, without assuming symmetry, why time-independent endstates can be reached in black hole and collapse simulations, with and without excision. Generalizing earlier work in spherical symmetry, we characterize the Killing states of the Bona–Massó slicing condition with time derivative along the normals to the slice (‘BMn’) as solutions of a mixed elliptic/hyperbolic differential equation on the slice. We show numerically in spherical symmetry that these steady states can be reached as end states from typical initial data with excision but can be reached with the puncture method only if the puncture is not numerically well resolved. As pointed out in [19] the moving puncture method typically uses a shift that systematically underresolves the puncture, and thus this method becomes a type of excision. During the evolution, BMn slicings often form gauge shocks. It may be that these are not seen in current 3D simulations only through the systematic underresolution provided by the shift, although we expect that they can be avoided with some care. Finally we point out that excision with BMn as currently implemented is ill-posed and therefore not expected to converge; this can be cured. In technical appendices, we derive the equations of pure gauge systems on a fixed spacetime, and bring the BSSN/NOR equations into three-dimensional tensor form suitable for multiple coordinate patches or spherical polar coordinates.

Initial value formalism for Lemaitre-Tolman-Bondi collapse

Formulating a dust-filled spherically symmetric metric utilizing the 3 + 1 formalism for general relativity, we show that the metric coefficients are completely determined by the matter distribution throughout the spacetime. Furthermore, the metric describes both inhomogeneous dust regions and also vacuum regions in a single coordinate patch, thus alleviating the need for complicated matching schemes at the interfaces. In this way, the system is established as an initial boundary value problem, which has many benefits for its numerical evolution. We show the dust part of the metric is equivalent to the class of Lemaitre-Tolman-Bondi (LTB) metrics under a coordinate transformation. In this coordinate system, shell crossing singularities (SCS) are exhibited as fluid shock waves, and we therefore discuss possibilities for the dynamical extension of shell crossings through the initial point of formation by borrowing methods from classical fluid dynamics. This paper fills a void in the present literature associated with these collapse models by fully developing the formalism in great detail. Furthermore, the applications provide examples of the benefits of the present model.

Numerical method for binary black hole/neutron star initial data: Code test

A new numerical method to construct binary black hole/neutron star initial data is presented. The method uses three spherical coordinate patches; two of these are centered at the binary compact objects and cover a neighborhood of each object; the third patch extends to the asymptotic region. As in the Komatsu-Eriguchi-Hachisu method, nonlinear elliptic field equations are decomposed into a flat space Laplacian and a remaining nonlinear expression that serves in each iteration as an effective source. The equations are solved iteratively, integrating a Green's function against the effective source at each iteration. Detailed convergence tests for the essential part of the code are performed for a few types of selected Green's functions to treat different boundary conditions. Numerical computation of the gravitational potential of a fluid source, and a toy model for a binary black hole field, are carefully calibrated with the analytic solutions to examine accuracy and convergence of the new code. As an example of the application of the code, an initial data set for binary black holes in the Isenberg-Wilson-Mathews formulation is presented, in which the apparent horizons are located using a method described in Appendix 1.